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Their paper also argued that a modest number of receptors spread over the surface captures most of the possible signal, and bacteria such as E. coli sense close to this physical limit.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eδc \/ c ~ 1 \/ √(D a c T)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eδc\/c: fractional error in concentration\u003c\/li\u003e\n\u003cli\u003eD: diffusion coefficient (m²\/s)\u003c\/li\u003e\n\u003cli\u003ea: cell radius (m)\u003c\/li\u003e\n\u003cli\u003ec: concentration (1\/m³)\u003c\/li\u003e\n\u003cli\u003eT: averaging time (s)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eBerg \u0026amp; Purcell, “Physics of chemoreception”, Biophysical Journal 20, 193 (1977).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e This is a scaling law (“~”, not “=”); the careful “perfect instrument” result adds a numerical factor. 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It slides down a funnel and drops into its fold.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eCyrus Levinthal pointed out in a 1969 talk that a protein cannot find its folded shape by random search. Take illustrative numbers: 100 residues, 3 shapes each, 10⁻¹³ s per try. That is 3¹⁰⁰ ≈ 5 × 10⁴⁷ shapes, and a blind search would take about 10²⁷ years. Real proteins fold in milliseconds to seconds.\u003c\/p\u003e\u003cp\u003eLevinthal's own point was that folding must follow pathways. The modern picture is a funnel-shaped energy landscape that biases the chain downhill, so it never samples most states. 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The record climbs year on year, with a seasonal zigzag as northern plants take up and release CO₂.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e315.98 → 427.35 ppm\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\u003cli\u003eppm: parts per million, mole fraction in dry air\u003c\/li\u003e\u003c\/ul\u003e\u003cp\u003eNOAA annual means: 315.98 ppm (1959). 427.35 ppm (2025, preliminary).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The 2025 annual mean is NOAA's preliminary value, ±0.12 ppm, and may be revised.\u003c\/p\u003e\u003ch3\u003eSources\u003c\/h3\u003e\u003cul\u003e\n\u003cli\u003eNOAA GML, Mauna Loa annual mean CO₂, gml.noaa.gov\/ccgg\/trends (co2_annmean_mlo.txt).\u003c\/li\u003e\n\u003cli\u003eC. D. 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In light, the vanes spin with the dark sides retreating. If light pressure drove it, the reflective sides would be pushed harder (a reflected photon gives twice the push of an absorbed one) and it would spin the other way. Instead, the dark side gets warmer, and the thin gas does the work. James Clerk Maxwell (Phil. Trans. R. Soc. 170, 231, 1879) and Osborne Reynolds, in his study of thermal transpiration (Phil. Trans. R. 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Even at absolute zero the dilute phase keeps a few percent of ³He dissolved. Ifft, Edwards and Sarwinski measured this limit at 6.37 ± 0.05% (Phys. Rev. Lett. 19, 831, 1967); later work gives about 6.4–6.6% depending on pressure. Pumping ³He out of the dilute phase pulls more across the boundary, and that ‘evaporation’ into the dilute phase absorbs heat. It works continuously, down to a few millikelvin.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ex\u003csub\u003e3\u003c\/sub\u003e(T → 0) ≈ 6.4–6.6%\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\u003cli\u003ex\u003csub\u003e3\u003c\/sub\u003e: fraction of ³He atoms in the dilute (⁴He-rich) phase\u003c\/li\u003e\u003c\/ul\u003e\u003cp\u003eThis is not a compression cycle: nothing is squeezed. 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Friction of the paddles warmed the water a tiny amount. Dividing the work done by the falling weights by the heat produced, he reported in 1850 (“On the Mechanical Equivalent of Heat”, Phil. Trans. R. Soc. 140, 61) that 772 foot-pounds of work raise one pound of water by 1 °F.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eW = J Q\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eW: mechanical work (ft·lbf)\u003c\/li\u003e\n\u003cli\u003eJ: mechanical equivalent of heat (ft·lbf per BTU)\u003c\/li\u003e\n\u003cli\u003eQ: heat produced (BTU)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eHeat and work are two forms of one thing, energy: the first law of thermodynamics. 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John B. Johnson measured it at Bell Labs and Harry Nyquist explained it, both in 1928 (Phys. Rev. 32, 97 and 32, 110). The noise power per unit bandwidth depends only on temperature and resistance, not on the material.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e⟨V²⟩ = 4k\u003csub\u003eB\u003c\/sub\u003eTRΔf\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003e⟨V²⟩: mean-square noise voltage (V²)\u003c\/li\u003e\n\u003cli\u003ek\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, exactly 1.380 649 × 10⁻²³ J\/K since 2019\u003c\/li\u003e\n\u003cli\u003eT: temperature (K)\u003c\/li\u003e\n\u003cli\u003eR: resistance (Ω)\u003c\/li\u003e\n\u003cli\u003eΔf: measurement bandwidth (Hz)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eA 1 kΩ resistor at room temperature gives about 4 nV per √Hz. Because k\u003csub\u003eB\u003c\/sub\u003e is now exact, Johnson noise can be used as a primary thermometer.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The formula is the low-frequency limit; at very high frequency or low temperature quantum corrections (hf ≳ k\u003csub\u003eB\u003c\/sub\u003eT) cut it off.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eNyquista holds a striped resistor with jagged orange lightning crackling from it. Printed text: \u003cstrong\u003eNYQUISTA\u003c\/strong\u003e with the line \u003cstrong\u003e4kTR\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50964270022759,"sku":"18220938352407663537","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50964270055527,"sku":"13187252428219160049","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50964270088295,"sku":"88588922017894369528","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50964270121063,"sku":"22322011856768034992","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/12247293452936607359_1200.jpg?v=1793663983"},{"product_id":"calorus-pixel-hoodie","title":"Calorus – The Caloric Theory of Heat Pixel Hoodie","description":"\u003cp\u003eCalorus, a bronze-armoured knight with a sealed glass chest-plate of glowing fluid, is pierced by a blunt borer; from the wound pours heat-shimmer, not fluid, and the vessel stays full.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eAntoine Lavoisier’s \u003cem\u003eTraité élémentaire de chimie\u003c\/em\u003e (1789) listed caloric, a conserved, weightless heat fluid, among the elements. In 1798 Count Rumford reported that boring a cannon with a blunt borer kept producing heat seemingly without limit, enough to boil water in about two and a half hours, and argued that heat is a form of motion. The fluid picture faded around 1850, when J. P. Joule’s “On the mechanical equivalent of heat” and Rudolf Clausius’s founding paper of thermodynamics made heat a transfer of energy, not a substance.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔU = Q − W\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eJoule 1850: 772 ft·lbf per Btu  ≈ 4.15 J per calorie   (modern: 1 cal\u003csub\u003eth\u003c\/sub\u003e = 4.184 J exactly)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eΔU: change in a system’s internal energy (J)\u003c\/li\u003e\n\u003cli\u003eQ: heat added to the system (J)\u003c\/li\u003e\n\u003cli\u003eW: work done by the system (J)\u003c\/li\u003e\n\u003cli\u003eft·lbf: foot-pound-force, ≈ 1.356 J\u003c\/li\u003e\n\u003cli\u003eBtu: heat to warm one pound of water by 1 °F, ≈ 1055 J\u003c\/li\u003e\n\u003cli\u003ecal\u003csub\u003eth\u003c\/sub\u003e: thermochemical calorie (J)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eThe first law says heat and work are two ways of moving the same energy, so there is no stock of caloric to run out of, and none to conserve. Rumford’s borer converted mechanical work W into internal energy indefinitely. Sources: Rumford, \u003cem\u003ePhil. Trans. R. Soc.\u003c\/em\u003e 88, 80 (1798), doi:10.1098\/rstl.1798.0006; Joule, \u003cem\u003ePhil. Trans. R. Soc.\u003c\/em\u003e 140, 61 (1850), doi:10.1098\/rstl.1850.0004; Clausius, \u003cem\u003eAnn. Phys.\u003c\/em\u003e 155, 368 (1850), doi:10.1002\/andp.18501550306.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Heat-fluid ideas are older than 1789; Lavoisier named and listed caloric. Rumford’s result was not accepted as decisive at the time, and 1850 is a conventional end date, marked by Joule’s and Clausius’s 1850 papers; caloric theory faded gradually.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eThey pierced me and waited for me to drain. I never did. Whatever pours out of me is motion, and the more they bore, the more I give.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the hoodie\u003c\/h3\u003e\u003cp\u003eFront: a cracked glass flask holding a flame, with a borer across it. Back: Calorus in full pixel art with the character name and printed dates for the theory (RUMFORD 1798 - JOULE 1850). Gildan 18500 Heavy Blend hoodie, Black, S to 5XL.\u003c\/p\u003e\u003cp\u003ePart of the \u003cstrong\u003eGhost Particles That Were Wrong\u003c\/strong\u003e set with \u003ca href=\"\/de\/products\/sir-aetheron-pixel-hoodie\"\u003eSir Aetheron\u003c\/a\u003e, \u003ca href=\"\/de\/products\/enrayne-pixel-hoodie\"\u003eEnrayne\u003c\/a\u003e, \u003ca href=\"\/de\/products\/diphantom-pixel-hoodie\"\u003eDiphantom\u003c\/a\u003e, \u003ca href=\"\/de\/products\/prematura-pixel-hoodie\"\u003ePrematura\u003c\/a\u003e, \u003ca href=\"\/de\/products\/thetarch-pixel-hoodie\"\u003eThetarch\u003c\/a\u003e. Browse all \u003ca href=\"\/de\/collections\/hoodies\"\u003ephysics hoodies\u003c\/a\u003e, more \u003ca href=\"\/de\/collections\/thermodynamics-t-shirts\"\u003ethermodynamics designs\u003c\/a\u003e, or the \u003ca href=\"\/de\/collections\/deep-cuts\"\u003eDeep Cuts\u003c\/a\u003e.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"Black \/ S","offer_id":50964467875943,"sku":"30415561719805009476","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ M","offer_id":50964467908711,"sku":"15448148543604733055","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ L","offer_id":50964467941479,"sku":"65091056722388620558","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ XL","offer_id":50964467974247,"sku":"76945071081196408436","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 2XL","offer_id":50964468007015,"sku":"25708564579182111490","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 3XL","offer_id":50964468039783,"sku":"29606816254305219352","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 4XL","offer_id":50964468072551,"sku":"31385764956232233152","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 5XL","offer_id":50964468105319,"sku":"13291385646835404039","price":64.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/03_H14_d3e5dd7f-b9ca-4288-847d-5206062addb0.png?v=1793678437"},{"product_id":"thermodynamics-sticker-sheet","title":"Thermodynamics Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Thermodynamics. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dilutra-dilution-refrigerator\"\u003eDilutra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eBelow about 0.87 K a mixture of helium-3 and helium-4 separates into a ³He-rich phase floating on a dilute phase. Even at absolute zero the dilute phase keeps a few percent of ³He dissolved. Ifft, Edwards and Sarwinski measured this limit at 6.37 ± 0.05% (Phys. Rev. Lett. 19, 831, 1967); later work gives about 6.4–6.6% depending on pressure. Pumping ³He out of the dilute phase pulls more across the boundary, and that ‘evaporation’ into the dilute phase absorbs heat. It works continuously, down to a few millikelvin.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ex\u003csub\u003e3\u003c\/sub\u003e(T → 0) ≈ 6.4–6.6%\u003c\/strong\u003e\u003cbr\u003ex\u003csub\u003e3\u003c\/sub\u003e: fraction of ³He atoms in the dilute (⁴He-rich) phase\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/boltzbrain-boltzmann-brain\"\u003eBoltzbrain\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the 1890s Ludwig Boltzmann suggested that our ordered world might be a rare fluctuation in a universe otherwise in thermal equilibrium. The difficulty is that a small fluctuation is far more likely than a big one. A single brain, complete with false memories, needs a much smaller drop in entropy than a whole ordered universe, so in such a picture most observers would be lone “Boltzmann brains”. Andreas Albrecht and Lorenzo Sorbo (Phys. Rev. D 70, 063528, 2004) discussed it as the so-called “Boltzmann's brain” paradox.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eP ∝ exp(ΔS\/k\u003csub\u003eB\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eP: relative probability of a fluctuation; ΔS: entropy change of the fluctuation, negative for a fluctuation into order (J\/K); k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, 1.380 649 × 10⁻²³ J\/K exactly in the 2019 SI\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/nyquista-johnson-noise\"\u003eNyquista\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAny resistor at a temperature above absolute zero hisses with voltage noise, because its electrons jiggle thermally. John B. Johnson measured it at Bell Labs and Harry Nyquist explained it, both in 1928 (Phys. Rev. 32, 97 and 32, 110). The noise power per unit bandwidth depends only on temperature and resistance, not on the material.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e⟨V²⟩ = 4k\u003csub\u003eB\u003c\/sub\u003eTRΔf\u003c\/strong\u003e\u003cbr\u003e⟨V²⟩: mean-square noise voltage (V²); k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, exactly 1.380 649 × 10⁻²³ J\/K since 2019; T: temperature (K); R: resistance (Ω); Δf: measurement bandwidth (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/seebecka-seebeck-effect\"\u003eSeebecka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1821 Thomas Johann Seebeck joined bismuth and copper in a loop and heated one junction. A compass needle beside it swung. A temperature difference between the junctions drives a voltage, and in a closed loop a current: thermoelectricity. He reported it to the Prussian Academy (Abh. Akad. Wiss. Berlin, 1822–23).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV = S ΔT\u003c\/strong\u003e\u003cbr\u003eV: thermoelectric voltage (V); S: Seebeck coefficient difference of the two metals (V\/K); ΔT: temperature difference between the junctions (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/invertia-negative-absolute-temperature-pixel-sticker\"\u003eInvertia\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTemperature is defined by how entropy changes when you add energy. In a system with a top energy level, like nuclear spins, putting more particles in the upper state than the lower one makes that slope negative: a negative absolute temperature, which is actually 'hotter than infinity'.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e1\/T = ∂S\/∂E \u0026lt; 0\u003c\/strong\u003e\u003cbr\u003eT: absolute temperature, in the standard (Boltzmann) definition (K); S: entropy (Boltzmann form, from the number of states at energy E) (J\/K); E: total energy of the spin system (J); N_↑, N_↓: number of particles in the upper and the lower level; Δ E: energy gap between the two levels (J); k_B: Boltzmann constant (J\/K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/paddlewheel-joule\"\u003ePaddlewheel\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eJames Prescott Joule let weights fall on pulleys, turning a paddle wheel inside an insulated vessel of water. Friction of the paddles warmed the water a tiny amount. Dividing the work done by the falling weights by the heat produced, he reported in 1850 (“On the Mechanical Equivalent of Heat”, Phil. Trans. R. Soc. 140, 61) that 772 foot-pounds of work raise one pound of water by 1 °F.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eW = J Q\u003c\/strong\u003e\u003cbr\u003eW: mechanical work (ft·lbf); J: mechanical equivalent of heat (ft·lbf per BTU); Q: heat produced (BTU)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/radiometra-crookes-radiometer\"\u003eRadiometra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA Crookes radiometer has four vanes, each black on one side and silver on the other, inside a bulb pumped down to a partial vacuum. In light, the vanes spin with the dark sides retreating. If light pressure drove it, the reflective sides would be pushed harder (a reflected photon gives twice the push of an absorbed one) and it would spin the other way. Instead, the dark side gets warmer, and the thin gas does the work. James Clerk Maxwell (Phil. Trans. R. Soc. 170, 231, 1879) and Osborne Reynolds, in his study of thermal transpiration (Phil. Trans. R. Soc. 170, 727, 1879), explained it: gas creeps around each vane's edge from the cold side to the hot side, and the reaction pushes the vane towards its cooler, silver side.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003edark side warmer ⇒ vane moves dark side trailing\u003c\/strong\u003e\u003cbr\u003edark side: absorbs light and heats up; silver side: reflects light and stays cooler; gas: thin enough for molecules to travel far between collisions, but not a hard vacuum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ratchetta-feynman-ratchet\"\u003eRatchetta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe ratchet and pawl, analysed by Smoluchowski (1912) and made famous in the Feynman Lectures (Vol. I, ch. 46), is a vane in a gas joined by an axle to a ratchet wheel with a spring-loaded pawl. Molecules hitting the vane seem to push the wheel forward, while the pawl blocks it going back. But the pawl is also at a temperature and also jiggles. When both are at the same temperature, the pawl lifts and lets the wheel slip back as often as the vane pushes it forward.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003erate forward ∝ e\u003csup\u003e−ε\/k\u003csub\u003eB\u003c\/sub\u003eT\u003csub\u003e1\u003c\/sub\u003e\u003c\/sup\u003e,   rate back ∝ e\u003csup\u003e−ε\/k\u003csub\u003eB\u003c\/sub\u003eT\u003csub\u003e2\u003c\/sub\u003e\u003c\/sup\u003e  ⇒  T\u003csub\u003e1\u003c\/sub\u003e = T\u003csub\u003e2\u003c\/sub\u003e gives ⟨W⟩ = 0\u003c\/strong\u003e\u003cbr\u003eε: energy needed to lift the pawl over one tooth (J); k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, exact in the 2019 SI; T\u003csub\u003e1\u003c\/sub\u003e: temperature of the gas around the vane (K); T\u003csub\u003e2\u003c\/sub\u003e: temperature of the ratchet and pawl (K); ⟨W⟩: average work delivered per unit time\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ruchardta-ruchardt-experiment\"\u003eRuchardta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1929 Eduard Rüchardt measured the heat-capacity ratio of a gas with a simple trick (Phys. Z. 30, 58). A ball fits snugly in the neck of a large flask. Push it and it bounces up and down on the springy gas below. The squeezes are quick, so no heat flows and the gas acts adiabatically. Timing the oscillation gives γ.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eγ = 4π²mV\/(A²pT²)\u003c\/strong\u003e\u003cbr\u003eγ: ratio of heat capacities C\u003csub\u003ep\u003c\/sub\u003e\/C\u003csub\u003ev\u003c\/sub\u003e; m: mass of the ball (kg); V: volume of gas (m³); A: cross-section of the neck (m²); p: gas pressure under the ball (Pa); T: oscillation period (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/sortex-maxwells-demon-pixel-sticker\"\u003eSortex\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLandauer argued that erasing one bit of information must release at least k_B T ln 2 of heat. Because a Maxwell's demon has to erase its records eventually, this cost is what saves the second law of thermodynamics.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ\u003csub\u003eerase\u003c\/sub\u003e ≥ k\u003csub\u003eB\u003c\/sub\u003eT ln 2\u003c\/strong\u003e\u003cbr\u003eQ_(erase): heat released to the surroundings when one bit of memory is erased (a lower bound, not a fixed cost) (J); k_B: Boltzmann constant (J\/K); T: temperature of the surroundings that absorb the heat (K); ln 2: natural logarithm of 2: one bit is a choice between two states\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/argonix-rayleigh-ramsay\"\u003eArgonix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLord Rayleigh weighed nitrogen with great care and found that nitrogen taken from air was about half a percent denser than nitrogen made from ammonia, a gap far larger than his errors. With William Ramsay he removed all the nitrogen and oxygen from air and found a stubborn leftover gas, about 1% of the atmosphere, with a new spectrum and no chemistry at all. They named it argon, ‘the lazy one’ (“Argon, a New Constituent of the Atmosphere”, Phil. Trans. A 186, 187, 1895).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eρ\u003csub\u003eair N₂\u003c\/sub\u003e ≈ 1.005 ρ\u003csub\u003echem N₂\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eρ\u003csub\u003eair N₂\u003c\/sub\u003e: density of ‘nitrogen’ from air, which still contains argon; ρ\u003csub\u003echem N₂\u003c\/sub\u003e: density of nitrogen from chemical sources\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/echobel-acoustic-gas-thermometer\"\u003eEchobel\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn acoustic gas thermometry, the resonant frequencies of a gas-filled spherical cavity of known size give the speed of sound. For a dilute monatomic gas the speed of sound fixes k\u003csub\u003eB\u003c\/sub\u003eT per atom. Measurements of this kind at the triple point of water (NPL, LNE, INRiM and others) gave the most precise values of the Boltzmann constant before the 2019 redefinition.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ec² = γk\u003csub\u003eB\u003c\/sub\u003eT\/m\u003c\/strong\u003e\u003cbr\u003ec: speed of sound in the zero-pressure limit (m\/s); γ: heat-capacity ratio, 5\/3 for an ideal monatomic gas; k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, 1.380 649 × 10⁻²³ J\/K exactly; T: thermodynamic temperature (K); m: mass of one gas atom (kg)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966901850215,"sku":"29053796791254063365","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/7530443038865545052_2048.jpg?v=1793676438"}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/collections\/thermodynamics-t-shirts.png?v=1793679027","url":"https:\/\/pixelatedphysics.com\/de\/collections\/thermodynamics-t-shirts.oembed?page=2","provider":"Pixelated Physics","version":"1.0","type":"link"}